Properties for internal stability of a discrete-time system

Properties for internal stability of a discrete-time system

These are two parts of a larger proof I'm working on, can't figure how i)
implies ii) though.
Dynamic system:
$x_{(k+1)} = Ax_{k}, x(0)=x_0$
Where $A \in \mathbb{R}^{n\times n} $ is a real constant matrix.
Properties:
i) All the eigenvalues of A are located on the open unit disc
ii) $\bigvee x_{0} \in \mathbb{R}^{n}, \sum\limits_{k=0}^\infty
||x_{k}||^2 < +\infty$
I understand that the eigenvalues are bounded in the disc, and that ii)
shows the system is finite and won't blow up, but I honestly just can't
find a way to show that i) implies ii). Here are some other tidbits I
have:
$x(k) = A^kx_{0}$
$A^k \backsim e^{At}$
$||A^k|| \leqq \beta |\lambda|^k$
$||e^{At}|| \leqq \beta e^{-\alpha t}$